Dijkgraaf-Witten theory


\infty-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory




Quantum field theory


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Dijkgraaf-Witten theory in dimension nn is the topological sigma-model quantum field theory whose target space is the classifying space of a discrete group and whose background gauge field is a circle n-bundle with connection on BG\mathbf{B}G (necessarily flat) which is equivalently a cocycle in the group cohomology of GG with coefficients in the circle group.

Viewed in a broader context and generalizing: Dijkgraaf-Witten theory is the ∞-Chern-Simons theory induced from a characteristic class c:BGB nU(1)\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n U(1) on a discrete ∞-groupoid BG:=DiscBG\mathbf{B}G := Disc B G. If GG here is an ordinary discrete group this is traditional Dijkgraaf-Witten theory, if GG is a discrete 2-group and the background field is a circle 4-bundle, then this is called the Yetter model.

This are the first two steps in filtering of target spaces by homotopy type truncation of ∞-Chern-Simons theory with discrete target spaces.

Concise survey of the ingredients

We may think of this as describing the quantum mechanics of an (n1)(n-1)-brane with nn-dimensional worldvolume Σ\Sigma propagating on BGB G and being charged under a higher analog of the electromagnetic field:

a field configuration over Σ\Sigma (a Σ\Sigma-shaped trajectory) is a morphism ϕ:ΣBG\phi : \Sigma \to \mathbf{B}G, hence equivalently a GG-principal bundle on Σ\Sigma. The configuration space of fields over Σ\Sigma is the groupoid of GG-principal bundles over Σ\Sigma.

The background gauge field is a morphism α:BGB nU(1)\alpha : \mathbf{B}G \to \mathbf{B}^n U(1) – hence a characteristic class for GG: a cocycle of degree nn in the group cohomology of GG.

The value of the Lagrangian L(ϕ)L(\phi) on a field configuration ϕ\phi is the characteristic class of this bundle with respect to the universal characteristic class of the given circle n-bundle:

L:(ΣϕBG)(α(ϕ):ΣϕBGαB nU(1)). L : (\Sigma \stackrel{\phi}{\to} \mathbf{B}G) \mapsto (\alpha(\phi) : \Sigma \stackrel{\phi}{\to} \mathbf{B}G \stackrel{\alpha}{\to} \mathbf{B}^n U(1)).

This is the classical field theory input of the model. The extended quantum field theory defined by this is supposed to be a rule that assigns space of states to lower dimensional pieces of Σ\Sigma and to nn-dimensional Σ\Sigmas a propagator.

The space of states assigned to a Σ\Sigma of dimension nkn-k for kk \in \mathbb{N} is the k-groupoid of sections of the higher line bundle associated to the circle (n-k)-bundle τ Σα\tau_\Sigma \alpha obtained by transgression of α\alpha to the mapping space H(Σ,BG)\mathbf{H}(\Sigma, \mathbf{B}G).

The propagator on Σ\Sigma of dimension nn is given by the path integral computed with measure the groupoid cardinality of BG\mathbf{B}G and integral kernel given by the action functional

exp(iS()):GBund(Σ)U(1) \exp(i S(-)) : G Bund(\Sigma) \to U(1)

that sends a field ϕ\phi to the evaluation of α(ϕ)\alpha(\phi) on the fundamental class of Σ\Sigma

exp(iS(ϕ))= Σα(ϕ). \exp(i S(\phi)) = \int_\Sigma \alpha(\phi) \,.

Gentle exposition


Details of DW-theory as an extended TQFT

The Dijkgraaf-Witten model is an example of (fully) extended topological quantum field theory. Namely, the above data not only assign an element in U(1)U(1) to any closed nn-dimensional manifold, but also a vector space to any closed (n1)(n-1)-dimensional manifold, a 2-vector space to any closed (n2)(n-2) manifold, and so on, ending with an n-vector space assigned to the point. Also, manifolds with boundary corresponds to (higher) linear operators between these (higher) vector spaces. According to the cobordism hypothesis, the whole structure of the Dijkgraaf-Witten model as an fully extended TQFT is contained in the datum of the nn-Vector space it assigns to the point. This is the space of sections of the flat nn-vector bundle BGnVect\mathbf{B}G\to n Vect induced by the background field BGB nU(1)\mathbf{B}G\to \mathbf{B}^n U(1).

Finite Group Cohomology

Since the target space of Dijkgraaf-Witten theory is the delooping groupoid BG\mathbf{B}G of a group GG (internal to Set), any background field given by a morphism α:BGA\alpha : \mathbf{B}G \to A in ∞Grpd is a cocycle in the group cohomology of GG, as described there.

Here we have a finite (or discrete) group GG, and a discrete abelian group AA, and we want to define H n(G;A)H^n(G;A). A way of doing this is to realize everything topologically: from GG we build the classifying space G\mathcal{B}G, and from AA the Eilenberg-MacLane space nA=K(A,n)\mathcal{B}^n A=K(A,n). Then we consider the space of maps hom(G, nA)hom(\mathcal{B}G,\mathcal{B}^n A) (these are our cocycles) and take its π 0\pi_0.

This way we have a familiar description, in a certain sense (topological spaces, continuous maps, homotopies,..), of the set H n(G;A)H^n(G;A). The drawback is that the topological spaces involved here are “gigantic” (infinite dimensional CW-complexes), where we had started with a very “little” datum: a finite group. So one can wonder if there is a finite model for the above construction, and the homotopy hypothesis serves it on a silver plate. Namely, since GG is discrete, G\mathcal{B}G is a 1-type, and nothing but the geometric realization of the delooping groupoid BG\mathbf{B}G (boldface BB here); similarly nA\mathcal{B}^n A is the topological geometric realization of the nn-groupoid B nA\mathbf{B}^n A, and the space of cocycles is hom(BG,B nA)hom(B G,B^n A). since GG is a finite group, BGB G is a finite groupoid, and so hom(BG,B nA)hom(B G,B^n A) is a finite set. This set is the finite model for hom(G, nA)hom(\mathcal{B}G,\mathcal{B}^n A) we were looking for.

To be continued…

The kk-vector spaces of states in codimension kk

The k-vector space associated with a closed oriented (nk)(n-k)-dimensional manifold X nkX_{n-k} admits a simple description in terms of sections:

The background field α:BGA\alpha : \mathbf{B}G \to A is transgressed to the mapping space [Π(X nk),BG][\Pi(X_{n-k}), \mathbf{B}G] by forming the internal hom

[Π(X nk),BG][Π(X nk),α][Π(X nk),A]τ kτ k[Π(X nk),A], [\Pi(X_{n-k}), \mathbf{B}G] \stackrel{[\Pi(X_{n-k}), \alpha]}{\to} [\Pi(X_{n-k}), A] \stackrel{\tau_{\leq k}}{\to} \tau_{\leq k} [\Pi(X_{n-k}), A] \,,

where the last morphism is the projection on the k-truncation. This defines a cocycle on the space of fields [Π(X nk),BG][\Pi(X_{n-k}), \mathbf{B}G] over X nkX_{n-k}, which classifies some principal ∞-bundle on this space. Given a canonical representation of the spaces of phases τ k[Π(X nk),A]\tau_k [\Pi(X_{n-k}), A] on a k-vector space we obtain the corresponding associated bundle over the space of fields. The (k1)(k-1)-category assigned by the extended topological quantum field theory to the closed X nkX_{n-k} is the category of sections of this kk-vector bundle.


We have

τ k[Π(X nk),B nU(1)]B kU(1) \tau_k [\Pi(X_{n-k}), \mathbf{B}^n U(1)] \simeq \mathbf{B}^k U(1)

By general abstract reasoning (recalled at cohomology and fiber sequence, for instance) we have for the homotopy groups that

π i[Π(X nk),B nU(1)]H ni(X nk,U(1)). \pi_i[\Pi(X_{n-k}),\mathbf{B}^n U(1)] \simeq H^{n-i}(X_{n-k}, U(1)) \,.

Now use the universal coefficient theorem, which asserts that we have an exact sequence

0Ext 1(H ni1(X nk,),U(1))H ni(X nk,U(1))Hom(H ni(X nk,),U(1))0. 0 \to Ext^1(H_{n-i-1}(X_{n-k},\mathbb{Z}),U(1)) \to H^{n-i}(X_{n-k},U(1)) \to Hom(H_{n-i}(X_{n-k},\mathbb{Z}),U(1)) \to 0 \,.

Since U(1)U(1) is an injective \mathbb{Z}-module we have

Ext 1(,U(1))=0. Ext^1(-,U(1))=0 \,.

Together this means that we have an isomorphism

H ni(X nk,U(1))Hom Ab(H ni(X nk,),U(1)) H^{n-i}(X_{n-k},U(1)) \simeq Hom_{Ab}(H_{n-i}(X_{n-k},\mathbb{Z}),U(1))

that identifies the cohomology group in question with the internal hom in Ab from the integral homology group of X nkX_{n-k} to U(1)U(1).

For i<ki\lt k, the right hand side is zero, and so

π i[Π(X nk),B nU(1)]=0fori<k. \pi_i[\Pi(X_{n-k}),\mathbf{B}^n U(1)]=0 \;\;\;\; for i\lt k \,.

For i=ki=k, instead, H ni(X nk,)H_{n-i}(X_{n-k},\mathbb{Z})\simeq \mathbb{Z}, since X nkX_{n-k} is a closed (nk)(n-k)-manifold and so

π k[Π(X nk),B nU(1)]U(1). \pi_k[\Pi(X_{n-k}),\mathbf{B}^n U(1)]\simeq U(1) \,.

Another proof of the isomorphism H nk(X nk,U(1))U(1)H^{n-k}(X_{n-k},U(1))\cong U(1) and of the identities H ni(X nk,U(1))=0H^{n-i}(X_{n-k},U(1))=0 for i<ki\lt k can be obtained as follows. Consider the short exact sequence of locally constant sheaves of abelian groups

0U(1)1. 0\to \mathbb{Z}\to \mathbb{R}\to U(1)\to 1.

This induces a long exact sequence in cohomology

H ni1(X nk,U(1))H ni(X nk,)H ni(X nk,)H ni(X nk,U(1))H ni+1(X nk,) \cdots \to H^{n-i-1}(X_{n-k},U(1))\to H^{n-i}(X_{n-k},\mathbb{Z})\to H^{n-i}(X_{n-k},\mathbb{R}) \to H^{n-i}(X_{n-k},U(1))\to H^{n-i+1}(X_{n-k},\mathbb{Z})\to \cdots

For i<ki\lt k we have H ni(X nk,U(1))=0H^{n-i}(X_{n-k},U(1))=0 by dimensional reasons, while for i=ki=k we find the exact sequence

H nk(X nk,)H nk(X nk,)H nk(X nk,U(1))0. \cdots \to H^{n-k}(X_{n-k},\mathbb{Z})\to H^{n-k}(X_{n-k},\mathbb{R}) \to H^{n-k}(X_{n-k},U(1))\to 0.

Since X nkX_{n-k} is a closed oriented manifold, we have H nk(X nk,)=H^{n-k}(X_{n-k},\mathbb{Z})=\mathbb{Z}, H nk(X nk,)=H^{n-k}(X_{n-k},\mathbb{R})=\mathbb{R}, and the map H nk(X nk,)H nk(X nk,)H^{n-k}(X_{n-k},\mathbb{Z})\to H^{n-k}(X_{n-k},\mathbb{R}) is the inclusion of \mathbb{Z} into \mathbb{R}. Hence H nk(X nk,U(1))/U(1)H^{n-k}(X_{n-k},U(1))\cong \mathbb{R}/\mathbb{Z}\cong U(1).

This means that the transgression of the Dijkgraaf-Witten background field

α:BGB nU(1) \alpha : \mathbf{B}G \to \mathbf{B}^n U(1)

to the space of field configurations [Π(X nk),BG][\Pi(X_{n-k}), \mathbf{B}G] over X nkX_{n-k} is a cocycle of the form

[Π(X nk),α]:[Π(X nk),BG]B kU(1). [\Pi(X_{n-k}), \alpha] : [\Pi(X_{n-k}), \mathbf{B}G] \to \mathbf{B}^k U(1) \,.

This classifies a B k1U(1)\mathbf{B}^{k-1} U(1)-principal ∞-bundle PP over the space of field configurations, given by the pullback

P EB k1U(1) [Π(X nk),BG] [Π(X nk),ρ] B kU(1). \array{ P &\to & \mathbf{E} \mathbf{B}^{k-1} U(1) \\ \downarrow && \downarrow \\ [\Pi(X_{n-k}), \mathbf{B}G] &\stackrel{[\Pi(X_{n-k}), \rho]}{\to}& \mathbf{B}^k U(1) } \,.

(Here EB k1U(1)\mathbf{E} \mathbf{B}^{k-1} U(1) is as described at universal principal ∞-bundle.)

By the canonical kk-representation ρ:B kU(1)kVect \rho : \mathbf{B}^k U(1) \to k Vect_{\mathbb{C}} of B k1U(1)\mathbf{B}^{k-1}U(1) on complex k-vector spaces, we have associated to this canonically a kk-vector bundle EE, which may be realized as the pullback

E kVect * [Π(X nk),BG] ρ[Π(X nk),ρ] kVect. \array{ E &\to & k Vect_* \\ \downarrow && \downarrow \\ [\Pi(X_{n-k}), \mathbf{B}G] &\stackrel{\rho \circ [\Pi(X_{n-k}), \rho]}{\to}& k Vect } \,.

Here kVect *k Vect_* is the k-category of pointed kk-vector bundles, see again generalized universal bundle for more.

If X nkX_{n-k} is closed then the kk-vector spaces associated by the TFT to X nkX_{n-k} is the (k-1)-category of sections of this bundle EE.

Relation to Chern-Simons theory

Dijkgraaf-Witten theory is to be thought of as the finite group version of Chern-Simons theory. Chern-Simons theory looks formally just as the above, only that all finite nn-groupoids appearing here are replaced by Lie ∞-groupoids (∞-stacks on CartSp).


The idea originates, of course, in

The discussion of the quasi-Hopf algebra associated with a group cohomology 3-cocycle c:BGB 3U(1)c \colon \mathbf{B}G \to \mathbf{B}^3 U(1) originates in

A review is in

and conceptual clarifications were established in

and, earlier, in an unpublished manuscript of Paul Bressler (2002-2004). See at Drinfeld double for more on this.

A first comprehensive structural account of DW theory as a functorial QFT was given in

A review is given on p. 68 of

First steps towards understand DW theory as an extended TQFT appear in

Discussion aiming towards a refinement of DW theory to an extended TQFT is in

Further conceptual refinement of this is indicated in section 3 and section 8 of

This proposes a general abstract way to construct path integral quantizations for finite group theories such as DW, see also at prequantum field theory. More along these lines is in

See also

For more on this see the discussion on the n-Forum.

Dijkgraaf–Witten theory via Kan extensions